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User blog:Ecl1psed276/Introducing a simple yet very powerful notation - Depth Notation!
In this blog post I will introduce the rules for my new notation called Depth Notation (which can be abbreviated as EDN - Ecl1psed's depth notation). I will also analyze it to some extent. Linear Depth Notation (LDN) The notation looks like this: D{a,b;c,d,e...} where there are exactly 2 entries before the semicolon and a nonnegative number of entries after the semicolon. Every entry must be a nonnegative integer. The first entry in the array is called the base, and the second entry is called the key. The ruleset for linear array notation is as follows: #Trailing zeros can be removed. For example, D{3,4;2,1,0} = D{3,4;2,1}. #If there are no entries after the semicolon, then the expression equals the sum of the base and the key. So D{9,13;} = 22. #If the key is 1, then the whole expression equals the base. So D{5,1;6,7} = 5. #If the third entry is not 0, then do this: ##Let X be the entire array, except with the key decremented by 1. ##Then, subtract 1 from the third entry, and replace the key with the value of X. For example, D{3,3;3,3,3} = D{3,D{3,2;3,3,3};2,3,3}. #If the third entry is 0, then find the first nonzero entry after the third entry. Subtract 1 from it, and replace the previous entry with the value of the key. For example, D{5,6;0,0,0,3,4} = D{5,6;0,0,6,2,4} Linear Depth Notation (LDN) acts very similarly to other linear array notations like SAN, BEAF, and BAN, so I will skim through the analysis pretty fast. The limit of LDN is \(\omega^\omega\). Analysis Dimensional Depth Notation (DDN) With Dimensional Depth Notation, we add new types of separators. The comma is a shorthand for (0), but now we can also have (1), (2), and we can have (n) for all natural numbers n. For example, D{3,3;3(5)8(123)4} is a valid array. The ruleset is as follows: #Trailing zeros can be removed. #If there are no entries after the semicolon, then the expression equals the sum of the base and the key. #If the key is 1, then the whole expression equals the base. #If the third entry is not 0, then do this: ##Let X be the entire array, except with the key decremented by 1. ##Then, subtract 1 from the third entry, and replace the key with the value of X. #If the third entry is 0, then find the first nonzero entry after the third entry. Call this entry y. #Look at the separator immediately before said entry. If it's (0), do this: ##Subtract 1 from the entry, and replace the previous entry with the key. #If the separator is (n) and n>0, then do this: ##Replace the 0(n)y with 0(n-1)0(n-1)0(n-1)...(n-1)0(n-1)1(n)y-1, the number of (n-1)'s is the key. For example, D{4,6;0(7)0(3)13,2} becomes D{4,5;0(7)0(2)0(2)0(2)0(2)0(2)1(3)12,2}. Analysis In this analysis, I will simplify things a little bit. Instead of writing out the FGH expression in full, I will just be giving the ordinal. Also, instead of writing the array out in full, like D{a,a;5,6,7}, I will only be giving the array part, which in this case is 5,6,7. Just like with the previous section, I will go decently fast through the analysis, because it is very similar to dimensional arrays in BEAF and BAN. So the limit of Dimensional Depth Notation is \(\omega^{\omega^\omega}\). Adding more stuff to EDN OK, now we have to cover a lot of new information about EDN before we can progress further. Get ready. Introducing Depth Now, I will introduce a new idea to you: Every separator and entry has a property called depth, which can be any nonnegative integer. Any entry on the base layer, and any separator on the base layer, has a depth of 0. If you have an entry within a separator, such as in the separator (3), then that entry has a depth of 1 more than the depth of the separator it's in. So in the expression D{a,b;3(5)3}, the 5 has a depth of 1, and the 3s each have a depth of 0. The separator (5) also has a depth of 0. Introducing Sets Now we will introduce another element to our arrays: the curly brackets { }. These are called sets. A set can contain an entire array, just like the () separators can. However, instead of just being a different type of separator, sets actually take the place of an entry in the array. For example, D{a,b;3({1,5})2} is a valid array which includes a set. The separator ({1,5}) contains only one entry, which is the set {1,5}. The set itself contains two entries, 1 and 5, and a separator, the comma. However, D{a,b;3(1{1,5})2} is not valid, because 1{1,5} just doesn't make sense. You need a separator between the 1 and the {1,5}, each of which is an entry in its own right. It is also important to note that trailing zeros can still be removed, even inside a seperator or a set. The last thing is that sets that contain only 1 number can be replaced with that number. For example, {5} is just equal to 5. And }}} also equals 5. However }}} cannot be simplified, since the set contains 2 numbers, 5 and 5. In this case, it is important to distinguish numbers and sets. If I said "sets that contain only 1 entry can be replaced with that number", then }}} would reduce, since the outer set only contains 1 entry (which is a set). But for this notation, }}} is quite different than }}}. Depth with Sets Ok, this will be a bit weird, but try to follow along. If you want to find the depth of an entry, you have to count each pair of () and {} it is inside. For every () it's in, you add 1 to the depth, but for every {} it's in, you have to subtract 1 ''from the depth. So if we have something like D{a,b;3({1,5})2}, then the 5 in the middle actually has a depth of 0, because it's inside one pair of () and one pair of {}. Negative depths are forbidden. So D{a,b;3( })2} is not a valid expression, because the depth of the 1 and 5 in the middle would be -2. Nesting Sets Interestingly, you can actually nest sets! For example, D{a,b;4(1( )2)3} is a valid array! The separator ( ) has a depth of 1, and the set has a depth of 2, since it's inside a depth 1 separator. The set {1,5} has a depth of 1, because with sets you have to subtract depth (explained above). And the innermost 1 and 5 each have a depth of 0. Note that you must make sure that no entry has a depth of less than 0, as explained above. Base Entries Let X be a set. If its first element is a number, then that number is the ''base entry of X. If its first element is a set, then that set's base entry will be the base entry of X. For example: The base entry of the set } is 4. The base entry of ,4},{1,2,3}}} is 5. An Example Let's talk about this expression: D{3,3;(1(6,{{7(2)10}(5)8})4)9} The 3's and the 9 have a depth of 0, since they are on the main array. The separator (1(6,{{7(2)10}(5)8})4) also has a depth of 0, since it's also on the main array. That separator has two entries, 1 and 4, along with the separator (6,{{7(2)10}(5)8}). This separator also has two entries, namely 6 and {{7(2)10}(5)8}, the latter of which is a set. This set has 2 entries, namely {7(2)10} and 8. It also has a separator, (5). The base entry of {{7(2)10}(5)8} is the 7. Finally, the set {7(2)10} contains the entries 7 and 10, and one separator, (2). How Sets Work Now we will see how sets actually work in the notation. Remember Rule 6 from the DDN section? It was the rule that is applicable if the first nonzero entry is immediately to the right of a (0). The rule says that we need to decrement that entry, and set the previous entry to the value of the key. However, that rule only applies on the base layer where the depth is 0. For higher depths, it acts differently. Basically, if we have an entry at a depth of N, we have to search out for the innermost separator whose depth is less than N. Then, we add a set and diagonalize. Examples The expression D{a,b({0,1})1} becomes D{a,b({b})1}, and then D{a,b(b)1} However, the expression D{a,b(0,1)1} is much more powerful, and it becomes D{a,b({0({0(...)1})1})1} with b nests. We can also say that D{a,b(0,1)1} becomes D{a,b( X->{0(X)1} )1} The expression D{a,b(0(0,1)1)1} becomes D{a,b(0({0({0({...})1})1})1)1} with b nests. The New Ruleset (COMING SOON!!!) Continuing with Analysis (Note: The ruleset is not formalized. I'm working on it.) (Another note: In the analysis, I will compare the ordinal level of individual seperators, not entire arrays. If a seperator S has ordinal level \(\alpha\), then the array D{a,b S 1} has a level of \(\omega^{\omega^\alpha}\) in the FGH. Up to the Bachmann-Howard Ordinal Up to \(\psi(\psi_I(0))\) Up to the Rathjen Ordinal Up to a compact The reason it took me so long to go past the Rathjen ordinal is because I couldn't figure out how the notation should work past it. But now I think I know. For the most part. But I don't entirely know, which is the reason why this analysis will go fast. Also, because the ordinal notations from here on out are very esoteric, I will not write out the ordinal in full, I will just write out the thing that needs to be collapsed using an appropriate collapsing function in order to produce the countable ordinal. For example, instead of writing \(\psi(\psi_{\chi(\varepsilon_{M+1})}(0))\), I would just say that you can collapse \(\chi(\varepsilon_{M+1})\). Anyway, let's continue with analysis. And beyond! (0(0,1)1) corresponds to M = {1{1,,2^,,}2} in pDAN. (0(0,2)1) corresponds to N = \(\Xi1\) = {1{1,,3^,,}2} in pDAN. (0(0,0,1)1) corresponds to K = {1{1,,1,,2^,,}2} in pDAN. (0(0,0,0,1)1) corresponds to {1{1,,1,,1,,2^,,}2}. (0(0(0,1)1)1) corresponds to {1{1{1,,2^,,}2^,,}2}. (0(0(0(0,1)1)1)1) corresponds to {1{1{1{1,,2^,,}2^,,}2^,,}2}. The limit of EDN (for now) is the same as the limit of pDAN. Category:Blog posts